Currently several different coherent optical sources exist that produce outputs in the mid infra-red region including, for example, quantum cascade lasers, infrared diode lasers, gas lasers notably CO2 and CO and optical parametric oscillators and optical parametric generators. For many of these systems the output is in the form of continuous emission or relatively long pulses in the picosecond or nanosecond range. Sources of sub-picosecond pulses in this wavelength range are relatively few and are restricted to complex systems based on synchronously pumped optical parametric oscillators (OPO) where the pump powers need not be very high, or optical parametric generators (OPG) which need to be pumped by very powerful ultra-short pulse lasers.
Nonlinear optical devices such as optical parametric amplifiers (OPA), optical parametric oscillators (OPO) and optical parametric generators (OPG) can all be based on three-wave mixing (3WM). These types of devices can be used to generate coherent mid infrared light. In such devices three waves at different optical frequencies interact in a second order nonlinear material, which is a material that displays a polarization quadratically proportional to the applied optical electric field. Only anisotropic materials display second order nonlinearity such as ferroelectric crystals. In the case of OPOs and OPAs it is conventional to designate the highest frequency wave the pump wave; the intermediate frequency wave the signal wave; and the lowest frequency wave the idler wave.
If the pump wave has an angular frequency ωp, the signal wave an angular frequency ωs and the idler wave an angular frequency ωi, in order to conserve energy the following relation holds: ωp=ωs+ωi. There is no particular restriction on the values ωp,s,i for energy conservation. However, for energy to transfer efficiently between these interacting waves a second condition must simultaneously be satisfied that represents momentum conservation. This is described by the relation kp=kski where kp,s,i are the wave-vectors of the interacting waves and kp,s,i=np,s,iωp,s,i/c where np,s,i are the linear refractive indices at the pump signal and idler frequencies respectively. Since, in general, np≠ns≠ni due to dispersion of the refractive index, then momentum conservation requires the use of special techniques. Several of these exist and include birefringent phase matching where the polarizations of the waves and the direction of propagation relative to the crystallographic axes in a birefringent crystal are chosen to achieve the phase matching condition; and quasi phase matching via periodic poling where the sign of the optical nonlinearity is reversed periodically along the propagation direction to achieve constructive interference of the generated signal and idler waves generated from different regions along the nonlinear crystal. In the latter case the periodic reversal of the sign of the nonlinearity creates an additional wave vector kg=2π/Λg where Λg is the period of the grating such that the phase matching relation becomes kp=ks+ki+kg. An appropriate change of Λg allows phase matching to be achieved.
The interaction between the waves is described by three coupled wave equations that can be found in many standard texts. It is worth noting that the energy conservation condition implies that destruction of a pump photon leads to creation of a pair of photons at the signal and idler frequencies. Thus energy is transferred from the pump wave to the signal wave and idler wave and this implies that the signal and idler waves may be amplified at the expense of the pump wave. In certain approximations, the gain at the signal and idler frequency can be represented by analytic expressions and again these can be found in standard texts.
When phase matched by using either birefringent phase matching or quasi phase matching, the total gain can be simplified as: G=¼ exp(2I) where  is the parametric gain coefficient, I is the propagation distance through the material and =sqrt(2ωsωideff2Ip/npnsniεoc3) where Ip is the pump intensity, deff is the effective second order nonlinearity of the material; εo is the permittivity of free space and np,s,i are the linear refractive indices at the pump, signal and idler respectively. It is worth noting that the gain of a parametric amplifier can be very large (>60 dB/cm) at intensities below the optical damage threshold of the material at least when using short optical pulses with duration less than a few picoseconds. In this respect optical parametric amplifiers can provide very much higher gain than common laser media.
The analytic expression for the gain presented above assumes the interaction occurs between continuous waves or relatively long pulses in the nonlinear crystal. However, as the pulses become shorter and, in particular, when they become shorter than about a picosecond, an additional factor must be taken into account.
In a homogeneous linear material the speed of propagation of a short pulse of less than a picosecond is determined by its group index, which at wavelength λ is determined by ng=n−λdn/dλ, where n is the linear refractive index of the medium at wavelength λ. Due to dispersion of the refractive index the group indices at different wavelengths are also different, which leads to pulses at different wavelengths propagating at difference group velocities given by vg=c/ng. The group velocity generally denotes the speed at which the peak of a pulse propagates and differs from the phase velocity of the wave fronts that make up the pulse that is determined directly by the refractive index, n. In addition, the pulses spread in time due to group velocity dispersion (GVD), where GVD=−λ/c d2n/d λ2. The values of both these parameters can vary significantly with wavelength.
By way of illustration, consider the case of a parametric amplifier based on periodically poled lithium niobate (PPLN). If we choose a pump wavelength at 1.04 μm which is close to that available from common neodymium or Ytterbium lasers, and a signal wavelength at 1.407 μm, which combined would generate an idler at 4 μm. The group velocities of the pump, signal and idler are c/2.21004; c/2.18242; c/2.23573 respectively where c is the speed of light in vacuum which is approximately 3×108 m/s. If we consider the case of pulses 100 fs in duration, then the pump and signal will separate after a propagation distance of only 1.09 mm in the crystal and the pump and idler after 1.17 mm. Thus, 3WM is usually limited only to mm lengths of the crystal at least for collinear propagation. Since the effective crystal length is very short this means that the total gain will be small unless extremely high laser intensity is used. The latter situation can be achieved in optical parametric generators pumped by very powerful short pulse lasers (>109 W) such as femtosecond amplified titanium sapphire lasers. Several commercial examples of this technology exist; however, these systems are both costly and complex and produce output pulses with very high peak powers which are unsuitable for many applications.
If the intensity remains low or moderate, the gain drops to a few dB/cm and this is only sufficient to create a so-called synchronously-pumped OPO. In this device, a pump laser generating short pulses at a high repetition rate (typically ≈50 MHz) pumps a short amplifying crystal contained within its own optical resonator which circulates pulses at either the signal or idler frequency (or both). The pulses return to the nonlinear crystal periodically and are amplified by the synchronous pump pulses, but only if the round trip time of the OPO resonator is carefully matched to the round trip time in the pump oscillator resonator. In such round trip matching conditions, pulses at the signal or idler wavelength can be amplified in successive transits of the OPO cavity and power in the signal and idler waves grows from noise. As this power grows, energy is transferred from the pump wave to the signal and idler waves. However, the round trip times of the two resonators (laser and OPA) must exactly match and the effects of group velocity dispersion between the interacting waves must be minimized, which in general means that the OPO crystal must be short compared with the length required for separation of the pulses due to the difference in their group velocities. Generally the cavity lengths must also be identical at the micron scale over distances of a meter or so. However, maintaining this length match in the face of mechanical or temperature fluctuations is challenging and requires complex locking schemes to obtain stable operation. Group velocity dispersion can be reduced by making the interacting beams non-collinear in the OPO crystal potentially allowing the use of longer crystals; however, the beams must then be large enough to maintain spatial overlap over the crystal length. Generating sub-picosecond sources in the mid infrared by a synchronously pumped OPO involves a high degree of complexity and expensive hardware. Nevertheless, these devices have applications in science and technology for probing vibrational states of molecules or as sources for nonlinear optics and commercial systems based on this principle are available.